In this note we characterize σ \sigma -finite Riesz measures that allow one to approximate measurable functions by continuous functions in the sense of Lusin’s theorem. We call such measures Lusin measures and show that not all σ \sigma -finite measures are Lusin measures. It is shown that if a topological space X X is either normal or countably paracompact, then every measure on X X is a Lusin measure. A counterexample is given to show that these sufficient conditions are not necessary.